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Lagrangian mechanics of fractional order, Hamilton-Jacobi fractional PDE and Taylor’s series of nondifferentiable functions. (English) Zbl 1154.70011

Summary: The paper proposes an extension of the Lagrange analytical mechanics to deal with dynamics of fractal nature. First of all, by using fractional difference, one introduces a slight modification of the Riemann-Liouville derivative definition, which is more consistent with self-similarity by removing the effect of initial value, and then for the convenience of the reader, one gives a brief background on Taylor’s series of fractional order \(f(x+h)=E_{\alpha}(h^{\alpha}D^{\alpha}_x)f(x)\) of nondifferentiable function, where \(E_{\alpha }\) is the Mittag-Leffler function. The Lagrange method of characteristics is extended for solving a class of nonlinear fractional partial differential equations. All this material is necessary to solve the problem of fractional optimal control and mainly to find the characteristics of its fractional Hamilton-Jacobi equation, therefore the canonical equations of optimality. Then fractional Lagrangian mechanics is considered as an application of fractional optimal control. In this framework, the use of complex-valued variables, as L. Nottale [ibid. 10, No. 2–3, 459–468 (1999; Zbl 0997.81526)] did it, appears as a direct consequence of the irreversibility of time.

MSC:

70Q05 Control of mechanical systems
70H03 Lagrange’s equations
70H20 Hamilton-Jacobi equations in mechanics
26A33 Fractional derivatives and integrals

Citations:

Zbl 0997.81526
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